Isogeometric Scaled Boundary Finite Element Method In Heat Transfer And Seepage Flow Problems And The Application To Dams

Since the heat conduction,thermal stress and seepage problems have been proven extremely important in water projects and many practical engineering applications,precise modelling and efficient simulation of these problems signify a critical issue in engineering techniques.Moreover,with the rapid improvement of the modern industrial design level,the geometry of structures used in practical engineering applications present considerable complexity.For the complex problems,the range of analogy solutions is extremely limited.For this reason,more and more numerical methods are proposed for simulation calculation.As a novel unconventional numerical technique,isogeometric scaled boundary finite element method(IGA-SBFEM)not only has the advantages of the conventional SBFEM,such as the semi-analytical solutions,reduction of spatial dimension by one,and no fundamental solution,but also puts the analysis of problems on the exact geometric models without losing any geometric accuracy.It eliminates the inconsistency between the geometric model of boundary and the analysis model in the conventional SBFEM,omitting the time-consuming process of the mesh generation.By introducting the Non-Uniform Rational B-splines(NURBS)to the boundary,this technique retains the geometric continuity of the boundary of solution domain to the computational physical field within the framework of the scaled boundary formulations.The IGA-SBFEM simplifies the design and analysis process and improves the accuracy,efficiency and convergence of numerical solutions when compared to SBFEM.In addition,based on the advantages of boundary discretization leading to a reduction of spatial dimension by one within the boundary technique,the proposed method not only fits neatly with the Boundary Representations(B-Reps)technique provided completely by Computer Aided Design(CAD),but also reduces the dependence of tensor-product form for NURBS in the traditional Isogeometric Analysis(IGA).Thus,the flexibility and effectiveness of dealing with complex geometric models have been greatly improved by the IGA-SBFEM.It combines the main advantages of SBFEM and IGA,and overcomes their respective disadvantages to a large extent.Based on the IGA-SBFEM,heat conduction,thermal stress and seepage problems with the complex geometry are studied in this paper.The main research of this paper is as follows:1.The basic IGA formulations for solving steady-state heat conduction problems are derived.The validity and superiority of IGA in solving heat conduction problem are demonstrated by numerical examples.Then,the IGA is applied to analyze the heat conduction problem of practical arch dam.Due to the topologic limitation of tensor-product patch,the NURBS-based IGA is difficulty to treat the problem of multi-connected domain with the complex cross-section.To resolve this issue,new searching algorithm and reconstruction technique of trimmed elements are introduced,and IGA with trimming technique is proposed.By using background surface and NURBS trimming curves,any complex topology can be effectively handled.This method is extended to model the heat conduction in structure with complex boundary shape,multi-voids structure and gravity dam with corridor.2.The IGA-SBFEM is applied to solve the two-dimensional steady-state heat conduction problems with complex geometries,Robin boundary condition and the prescribed heat fluxes and temperatures on the side-faces.The computational results show that these complex heat problems can be more effectively handled by IGA-SBFEM with superior numerical accuracy,efficiency and convergence properties when compared to the SBFEM.Further,the IGA-SBFEM in combination with the modified precise time step integration method(MPTSIM)is developed to perform two-dimensional transient heat conduction analysis.The space domain is first discretized by using the IGA-SBFEM to obtain the ordinary differential equations with respect to time,and then the ordinary differential equations are solved using the MPTSIM to obtain the variations of temperature with time step by step.Based on the basic equations of IGA-SBFEM,the solution implementation of MPTSIM for the transient heat conduction problems is derived.The boundary stiffness matrix is determined from the eigenvalue problem,and the mass matrix is obtained from the low-frequency approximation.The numerical results show that the proposed method not only has the numerical properties of high stability and accuracy,but also circumvents the numerical fluctuations in the conventional finite time difference method.3.Based on the accurate solutions of the temperature field in computational domain obtained using IGA-SBFEM.the method is further applied to solve the thermal stress problem.By taking the initial strains caused by the thermal deformation into the governing equations of the two-dimensional elastostatics,the non-homogeneous isogeometric scaled boundary finite element equation in displacement with the thermal loading can be derived by using the scaled boundary technique of thevector form.As the temperature change is expressed as a series of the power functions in the radial coordinate,the integral in particular solutions of the thermal loading can be evaluated analytically through the IGA-SBFEM.The final thermal stress field,which is analytical in the radial coordinate,can be obtained through the superposition of the solutions for the individual terms of the temperature series.The thermal stress field of a typical gravity dam is calculated by this method,and the applicability of IGA-SBFEM in practical engineering is proved.4.The IGA-SBFEM is presented for the solution of the steady-state axisymmetric heat conduction problems.And a new formulation is derived for the axisymmetric heat conduction problems considering the Robin boundary condition.Since the axisymmetric problems can be treated as the two-dimensional problems and only the boundary requires discretization with NURBS elements leading to a reduction of spatial dimension by one in the IGA-SBFEM,the axisymmetric problems can be further reduced to one dimension,while the rigid trivariate tensor product structure is also reduced to one dimension.No tensor-product structure in axisymmetric problems is required,which improves the flexibility for describing complex topology.Moreover,with the NURBS basis functions which can describe any free-form curves,the IGA-SBFEM not only eliminates the geometrical error of the boundary of meridional section,but also improves the accuracy and efficiency for solving the axisymmetric heat conduction problems.5.Based on the infinite solution procedure of IGA-SBFEM and the modified IGA-SBFEM for for semi-infinite problems with parallel side-faces,the IGA-SBFEM is developed for the analysis of complex steady seepage problems in unbounded domain and a band-shaped semi-infinite domain with parallel side-faces in multilayered soils.By illustrating numerical examples,the effectiveness and flexibility of the present approach for solving seepage problems involving the layered soils,multi-connected domains,and inclusions are demonstrated.In addition,both the confined and unconfined seepage flow can be effectively handled by IGA-SBFEM considering the unbounded domain and multilayered soils that consist of semi-infinite layers with a constant depth.By solving the complex seepage problems in dam,it is proved that the IGA-SBFEM can accurately calculate the final location of of free surface in dam and the distribution of underground seepage,thus the obtained results of IGA-SBFEM are very useful to master the seepage law of the complex seepage problem in dam.